CHAPTER 17 ENGINEERING COST ANALYSIS

Transcription

1 CHAPTER 17 ENGINEERING COST ANALYSIS Charles V. Higbee Geo-Heat Center Klamath Falls, OR INTRODUCTION In the early 1970s, life cycle costing (LCC) was adopted by the federal government. LCC is a method of evaluating all the costs associated with acquisition, construction and operation of a project. LCC was designed to minimize costs of major projects, not only in consideration of acquisition and construction, but especially to emphasize the reduction of operation and maintenance costs during the project life. Authors of engineering economics texts have been very reluctant and painfully slow to explain and deal with LCC. Many authors devote less than one page to the subject. The reason for this is that LCC has several major drawbacks. The first of these is that costs over the life of the project must be estimated based on some forecast, and forecasts have proven to be highly variable and frequently inaccurate. The second problem with LCC is that some life span must be selected over which to evaluate the project, and many projects, especially renewable energy projects, are expected to have an unlimited life (they are expected to live forever). The longer the life cycle, the more inaccurate annual costs become because of the inability to forecast accurately. This chapter on engineering cost analysis is designed to provide a basic understanding and the elementary skills to complete a preliminary LCC analysis of a proposed project. The time value of money is discussed and mathematical formulas for dealing with the cash flows of a project are derived. Methods of cost comparison are presented. Depreciation methods and depletion allowances are included combined with their effect on the after-tax cash flows. The computer program RELCOST, designed to perform LCC for renewable energy projects, is also presented. A discussion of caveats related to performing LCC is included. No one should attempt to do a comprehensive cost analysis of any project without an extensive background on the subject, and considerable expertise in the current tax law Use of Interest Tables When performing engineering cost analysis, it is necessary to apply the mathematical formulas developed in this chapter and avoid using interest tables for the following reasons: 1. Interest rates applying to real world problems are not found in interest tables, and therefore, interpolation is required. When trying to solve problems with interpolation the assumption is made that compound interest formulas are linear functions. THEY ARE NOT. They are logarithmic functions. 2. Not only are real world interest rates difficult to find in tables, but it is frequently difficult to find the required number of interest periods for the project in a set of tables. 3. If the need arises to convert a frequently compounded interest rate to a weekly or monthly interest rate, it is almost certain the value of the effective interest rate will not be in any interest table. 4. Renewable energy projects, especially those for district heating systems, can run into hundreds of millions of dollars. Although it is understood that this chapter was written for preliminary economic studies, nevertheless, interpolation of interest tables can cause an error many times larger than the cost analyst's annual salary. 5. With today's microcomputers and sophisticated handheld calculators, interest tables are obsolete. Calculators capable of computing all time value functions except gradients are available for under $20. These calculators can also solve the number of interest periods and iterate an interest rate to nine decimal places THE TIME VALUE OF MONEY The concept of the time value of money is as old as money itself. Money is an asset, the same as plant and equipment and other owned resources. If equipment is borrowed, a plant is rented or land is leased, the owner should receive equitable compensation for its use. If money is borrowed, the lender should be reimbursed for its use. The rent paid for using someone else's money is called interest. Interest takes two different forms: simple interest and compound interest. 359

2 Throughout this chapter, the time value of money and compound interest are used in the cost analysis of projects. Such things as risk and uncertainty are ignored, and the concept of an unstable dollar or the value of the dollar fluctuating in the foreign market are not considered. However, in LCC analysis of renewable energy projects, inflation rates for operation and maintenance, equipment purchases, energy consumed and the revenue from energy sold for both conventional and renewable energy, will be considered. The concept of the time value of money evolves from the fact that a dollar today is worth considerably more than a promise to pay a dollar at some future date. The reason this is true is because a dollar today could be invested and be earning interest such that, at sometime in the future, the interest earned would make the investment worth considerably more than one dollar. To illustrate the time value of money, it is convenient to consider money invested at a simple interest rate Simple Interest Simple interest is interest accumulated periodically on a principal sum of money that is provided as a loan or invested at some rate of interest (i), where i represents an interest rate per interest period. It is important to notice that in problems involving simple interest, interest is only charged or earned on the original amount borrowed or invested. Consider a deposit of $100 made into an account that pays 6% simple interest annually. If the money is left on deposit for 1 year, the balance at the end of year one would be: (100) = $106. If the money is left on deposit for 2 years, the balance at the end of year two would be: (100) (100) = $112. If the money is left on deposit for 3 years, the balance at the end of year three would be: (100) (100) (100) = $118. If n equals the number of interest periods the money is left on deposit and i equals the rate of interest for each period, the formula for calculating the balance at the end of n periods would be: (i x n). Substituting 6% for i and 3 for n, the formula becomes: (0.06 x 3) = $118. Substituting present value (Pv) for the amount of money loaned or deposited at time zero (beginning of the time period covered by the investment), and future value (Fv) for the balance in the account at the end of n periods, the formula becomes: Fv = Pv + Pv(i x n). Factoring out Pv, the formula becomes: Fv = Pv(1 + i x n). (17.1) Going back to the original values, if the $100 is left on deposit for 5 years, the future value would be $130, and is written: Pv = 100; i = 0.06; n = 5 Fv = 100( x 5) Fv = $130. Remember, in simple interest problems, interest is earned only on the amount of the original deposit. Consider the case where interest is calculated more frequently than once per year. This would not change the amount of money earned in simple interest calculations. Suppose that $100 was deposited for 5 years at a rate of 6% simple interest, calculated every three months. Since there are four 3-month periods in a year, the simple interest per interest period becomes 0.06/4 = and n becomes 4 quarters per year x 5 y = 20 total interest periods. Therefore: Fv = 100( x 20) Fv = $130. Applying this formula to the time value of money, it can be shown that for any given rate of interest, $100 received today would be much greater value than $100 received 5 years from today. Consider: Proposal 1: A promise to pay $100 5 years from today. Proposal 2: A promise to pay $100 today. If proposal 2 is accepted over proposal 1, the $100 received today could be deposited into an account that earned 9% annually, and in 5 years the balance would be $145. Using this same theory, the present value of a promise to pay $100 5 years from today can be evaluated as: Fv = 100; i = 0.09; n = 5 y 100 = Pv( x 5). Solving for Pv, the equation becomes: Pv = 100/( x 5) Pv = $

3 Throughout this chapter and in cost analysis texts in general, cash flow diagrams are normally drawn to illustrate monies flowing into or out of a project at some specific time period. The accepted convention is: a) money flowing out is indicated by a down arrow and b) money flowing in is indicated by an up arrow. Example 17.1: A $1,000 loan to be repaid in two equal annual payments, from the borrower's point of view, would be drawn as: and, from the lender's point of view, would be drawn as: shorthand method of representing a formula to be applied to a problem or a portion of a problem, rather than having to write the formula in its entirety. For example, (F/P, i, n) is read, "To find the future value F, given the present value, P at an interest rate per period i for n interest periods." This notation applies only to compound interest. Compound interest varies from simple interest in that interest is earned on the interest accumulated in the account. To illustrate: If $100 is deposited at 6% compound annually, at the end of the first year the balance would be: 100( ) = $106. This is the same as in simple interest. However, if the money is allowed to remain on deposit for 2 years, the interest earned during the second year would be: The examples below illustrate the application of cash flow diagrams. Example 17.2: A woman deposited $500 for 3 years at 7% simple interest per annum. How much money can be withdrawn from the account at the end of the 3-year period? The cash flow diagram below indicates money deposited into the investment as, and money withdrawn from the investment as, 106(0.06) = $6.36 giving a balance of $ at the end of the second year. If the money is left on deposit for 3 years, the interest earned during the third year would be: (0.06) = $6.74. Thus, the balance at the end of the third year would be: = $ solution: Fv = Pv(1 + i x n) Fv = 500( x 3) Fv = 500(1.21) Fv = $605. Example 17.3: Assume $500 is deposited for 200 days in an account that earns 6% simple interest per annum. What is the balance at the end of the investment period? The solution is: Fv = Pv(1 + i x n) Fv = 500[ (200/365)] Fv = 500(1.0329) Fv = $ Compound Interest All compound interest formulas developed will include the standard functional notation for those formulas to the right of the developed formula. Functional notation is a The mathematical function of compound interest for a deposit of $100 earning 6% compounded annually left on deposit for 3 years is stated and described mathematically below. Original deposit plus interest earned at the end of the first year becomes: Fv = 100( ) plus the interest earned during the second year: [100( )] plus the interest earned during the third year: +0.06{100( ) [100( )]}. The formula becomes rather complex with only a 3- year investment. The formula can be simplified through mathematical manipulation. For purposes of this illustration, let 0.06 = i and the number of interest periods = 3, then: Fv = 100(1 + i) + i[100(1 + i)] + i{100(1 + i) + i [100(1 + i)]}. 361

4 Factoring out $100 from the above equation: Fv = 100[(1 + i) + i(1 +i) + i{(1 + i) + i(1 + i)}] simplifying: Fv = 100[1 + i + i + i 2 + i(1 + i + i + i 2 )] simplifying further: Fv = 100(1 + i + i + i 2 + i + i 2 + i 2 + i 3 ) and collecting terms: Fv = 100(1 + 3i + 3i 2 + i 3 ) then, this equation can be factored into: Fv = 100[(1 + i)(1 + i)(1 + i)] = 100(1 + i) 3. Substituting n for the number of interest periods, which in this case is 3, the result is: Fv = 100(1 + i) n. Letting Pv = the amount of the investment, then: Fv = Pv(1 + i) n (F/P,i,n)(17.2) This is the single payment compound amount factor. Solving Equation (17.2) for Pv gives: (P/F,i,n)(17.3) which is the single payment present worth factor. With the development of the equation for finding the future value of a lump sum investment at a compound interest rate for n interest periods, it can be shown how more frequent compounding increases the interest earned. An interest rate of 3%/6 mo would be stated in nominal form as 6% compounded semiannually. Consider the following examples with interest rates stated as an annual percentage rate (APR), commonly referred to as the "nominal interest rate." Example 17.4: An amount of $100 is invested for 3 years in an account that earns 18% compounded annually. The future value at the end of the 3-year period will be: Fv = Pv(1 + i) n Fv = 100( ) 3 Fv = 100(1.6430) Fv = $ Example 17.5: Assume $100 is invested for three years in an account that earns 18% compounded quarterly. The future value at the end of the 3-year period is: where Solution: Fv = Pv(1 + i) n i = 0.18/4 quarters/y = per quarter n = 3 y x 4 quarters/y = 12 interest periods. Fv = 100( ) 12 Fv = 100(1.6959) Fv = $ Example 17.6: Suppose $100 is invested in an account that earns 18% compounded monthly. The future value at the end of a 3-year period is: where Solution: Fv = Pv(1 + i) n i = 0.18/12 months/y = 0.015/mo n = 3 y x 12 mo/y = 36 interest periods. Fv = 100( ) 36 Fv = 100(1.7091) Fv = $ Example 17.7: An amount of $100 is invested for 3 years in an account that earns 18% compounded weekly. The future value at the end of the 3-year period is: where Solution: Fv = Pv(1 + i) n i = 0.18/52 weeks/y = /week n = 3 y x 52 weeks/y = 156 weeks. Fv = 100( ) 156 Fv = 100(1.7144) Fv = $ Example 17.8: If $100 is invested for 3 years in an account that earns 18% compounded daily, the future value at the end of the 3-year period is: Fv = Pv(1 + i) n 362

5 where Solution: i = 0.18/365 days/y = /d n = 3 y x 365 d/y = 1095 d. Fv = 100( ) 1095 Fv = 100( ) Fv = $ Money invested today will grow to a larger amount in the future. If this is true, then the promise to pay some amount of money in the future is worth a smaller amount today. Example 17.9: What is the present value of a promise to pay $3,000 5 years from today if the interest rate is 12% compounded monthly? This can be written: where Solution: i = 0.12/12 = 0.01 n = 5 x 12 = 60. Pv = 3,000/( ) 60 Pv = $1, Annual Effective Interest Rates It is convenient at this point in the development of compound interest to introduce annual effective interest rates. Annual effective interest (AEI) is interest stated in terms of an annual rate compounded yearly, which is the equivalent of a nominally stated interest rate. Table 17.1 illustrates the relationship between nominal interest, interest rate per interest period, and annual effective interest. Notice that the nominal interest rate remains the same percentage while the compounding periods change. The interest rate per interest period is obtained by dividing the nominal rate by the number of interest periods per year. The annual effective interest rate is the only true indicator of the amount of annual interest, and therefore, annual effective interest provides a true measure for comparing interest rates when the frequency of compounding is different. The annual effective interest rate may be found for any nominal interest rate as shown below. Table 17.1 Comparative Interest Rates (APR) Nominal Interest Rate per Annual Effective Interest Rate Interest Period Interest (AEI) (18%) (%) (%) Compound annually Compounded semiannually Compounded quarterly Compounded monthly Compounded weekly Compounded daily Compounded continuously Consider a dollar that was invested for 1 year at a nominal rate of 18% compounded monthly. To calculate the balance (Fv) at the end of the year: therefore, Fv = 1[1 + (0.18/12)] 12 Fv = 1(1.015) 12 Fv = 1(1.1956) Fv = $ Because a dollar was invested originally, the annual interest earned may be found by subtracting the original investment: = and the effective interest is 19.56%. Then, the formula for finding annual effective interest is: (17.4) 363

6 where AEI = annual effective interest rate r = nominal interest rate/y m = number of interest periods/y. The annual percentage rate is divided by the number of compounding periods/y and raised to the power of the number of compounding periods/y, and 1 is subtracted from that answer to arrive at the annual effective interest rate. The following examples are used to further illustrate the differences between APR and AEI. Example 17.10: For an APR of 12% compounded annually, the annual effective interest (AEI) is 12%. Therefore: AEI = 12% compounded annually. There is no difference between the two. Example 17.11: For an APR of 12% compounded semi-annually, the semiannual effective interest rate is 0.12/2 = 0.06 or 6%, presented as: AEI = ( /2) 2-1 = or 12.36%. Example 17.12: For an APR of 12% compounded quarterly, the quarterly effective interest rate is 0.03 or 3%, AEI becomes: AEI = ( /4) 4-1 = or 12.55%. Example 17.13: For an APR of 12% compounded daily, the daily effective interest rate is 0.12/365 = or %, giving: AEI = ( /365) = or 12.75%. When interest is compounded continuously (when n approaches infinity), the annual effective interest rate takes the form of e - 1, where e = the natural logarithm Therefore, an APR of 12% compounded continuously would yield an AEI of ( ) = %. where r = annual percentage rate m = number of compound periods/year c = number of compound periods for the time frame of the effective interest rate. Example 17.14: The present value of a promise to pay $4,000 6 years from today at an interest rate that is compounded quarterly is $2,798. Find the nominal interest rate and the annual effective interest rate. Fv = Pv(1 + i) n where n = 6 x 4 = 24 quarters and solving 4,000 = 2,798(1 + i) 24 dividing both sides by 2, = (1 + i) 24 taking the 24th root of both sides ( ) = [(1 + i) 24 ] = 1 + i i = 0.015, or 1.5%. By definition, i is the interest rate per interest period. Therefore, the answer, 1.5%, is the interest rate per quarter. In order to find the nominal interest rate (or annual percentage rate), it is necessary to multiply i times the number of quarters per year. The nominal interest rate is x 4 = 0.06, or 6% compounded quarterly. The answer would be incorrect if the frequency of compounding was not included. If the nominal rate is given as 6%, this would indicate 6% compounded annually. The annual effective interest rate, using Equation (17.4), becomes: Effective interest rates can be calculated for periods other than annually. To find a weekly effective interest rate of 12% compounded daily, the weekly effective interest rate would be: AEI = ( ) 4-1 AEI = , or 6.14%. ( /365) 7-1 = , or 0.23%. Therefore, (17.5) 364

7 % % % 6% 4% 5 Figure Years Exponential nature of compound interest rates. The formulas developed for compound interest and the similar formula for converting APR to AEI are logarithmic functions (see Figure 17.1). When interest rates are extremely low, the number of compounding periods is almost insignificant. For example, 2% APR compounded annually is 2% AEI; 2% APR compounded daily is 2.02% AEI while 40% APR compounded daily jumps to 49.15% AEI. In evaluating projects for nonprofit organizations, interest rates are usually kept rather low, but there are many private entities that evaluate alternatives at the corporation s rate of return, which can be a very high rate ANNUITIES Ordinary Annuities The definition of an ordinary annuity is a stream of equal end-of-period payments. Ordinary annuities are the most common form of payment series used in cost analysis. Loan payments, car payments, charge card payments, maintenance and operating costs of equipment are all stated in terms of ordinary annuities. These payments are frequently monthly payments, but they can be weekly, yearly or any other uniform time period. The important thing is that they begin at the end of the first interest period. For example, a person purchases an automobile for $5,000 and is obligated to pay $125 per month for 48 months. The first payment will be due one month after the purchase of the car and the last payment will be due at the end of month 48. In order to derive a mathematical formula to evaluate ordinary annuities, it is convenient to use the future value formula for compound interest Equation (17.2). Consider three equal end-of-year payments of $100 each. To find the future value of this payment series at the end of year three, use the future value formula. Notice that the first payment is two interest periods before the end of the project. Therefore, the future value of the first payment is: Fv = Pv(1 + i) n Fv = 100(1 + i) 2. The future value of the second payment, which is one interest period before the end of the project, is: Fv = 100(1 + i) 1. The third payment is zero interest periods away from the end of the project. Therefore, the future value of payment number three is simply $100. The future value of all three payments is: Fv = 100(1 + i) (1 + i) Reversing the order of these terms and factoring out $100, the equation becomes: Fv = 100[1 + (1 + i) 1 + (1 + i) 2 ]. 365

8 Notice that with a 3-year project and three annual payments, n does not get higher than 2. This is because the payments are made at the end of each period. Letting A = the amount of each payment, a general equation can be written to find the future value of n payments as: Fv = A[1 + (1 + i) 1 + (1 + i) 2 + (1 + i) (1 + i) n-1 ]. Multiplying this equation by (1 + i) results in: Fv(1 + i) = A[(1 + i) 1 + (1 + i) 2 + (1 + I) 3 + (1 + i) (1 + i) n ]. Subtracting the first equation from the second equation: also and Fv(1 + i) - Fv = A[-1 + (1 + i) n ] Fv + i(fv) - Fv = A[(1 + i) n - 1] i(fv) = A[(1 + i) n - 1] dividing both sides of the equation by i gives: (F/A,i,n)(17.6) This is the uniform series compound amount factor. Solving for A instead of Fv gives: (A/F,i,n)(17.7) This the uniform series sinking fund factor. Returning to the future value formula, Fv = Pv(1 + i) n, and substituting the right side of this equation for Fv in Equation (17.6) gives: dividing both sides by (1 + i) n yields: (P/A,i,n)(17.8) This is the uniform series present worth factor. Solving the above equation for A instead of Pv gives: (A/P,i,n)(17.9) This is the uniform series capital recovery factor. Equation (17.9) is used for finding the payment series of a loan or the annual equivalent cost of a purchased piece of equipment. In calculating future value or present value of a payment series, i must equal the interest rate per payment period, and n must equal the total number of payments. If an interest rate is stated as 12% compounded monthly (APR), and the payment series is monthly, then i = 0.12/12 = 0.01 or 1%/mo. Complications can arise when the annual percentage rate is stated in such a manner as to be incompatible with the payment period. For example, assume a loan of $500 to be repaid in 26 equal end-of-week payments with an interest rate of 10% compounded daily. Before this problem can be solved, i must be stated in terms of 1 week. Therefore, the weekly effective interest rate, (WEI) must be calculated as: WEI = ( /365) 7-1 WEI = , or 0.19% per week. When interest is compounded more frequently than the payment, the amount of each payment (A) can be calculated in the above example by: A = 500(A/P,0.19,26) A = $ If the above problem had an interest rate of 10% compounded quarterly, this would present a case where interest is compounded less frequently than the payment period. Therefore, between compounding periods the interest rate is zero. The payment must coincide with the interest rate. Because 26 weeks constitutes 6 months, and the interest rate is compounded quarterly, there are two quarters in a 26-week period. Therefore, to find the interest rate per quarter, find the payment per quarter and divide that payment by 13 weeks to arrive at a payment per week. The interest rate per quarter is: 0.10/4 = or 2.5% A = 500(A/P,2.5,2) A = $259.41/quarter. Dividing the above answer by 13 gives $19.95 as the payment per week. Equation (17.8) is used to determine the present value of an ordinary annuity. 366

9 Returning to the problem at the beginning of Ordinary Annuities, $5,000 was financed on an automobile for 48 months at an interest rate of 9.24% compounded monthly. Find the monthly payment: However, the sixth payment does not exist. Therefore, that payment, which is zero interest periods away from Fv, must be subtracted from the above formula as: where Solution i = /12 = n = 4 x 12 = 48. A = 5,000(A/P,0.77,48) A = $ or $125/mo Annuities Due The definition of an annuity due is a stream of equal, beginning-of-the-period payments. Although this payment series is not nearly as common as an ordinary annuity, it is still found in many projects. Beginning-of-the-period payments apply to such things as rents, leases, insurance premiums, subscriptions, etc. Rather than attempt to derive a formula to evaluate annuities due, it is much simpler to modify the existing formulas that have already been derived. Consider a 5-year cash flow diagram with payments made at the beginning of each year. These payments will be designated as (a) to avoid confusion with an ordinary annuity. The diagram is shown as: Placing - 1 as the last value inside the brackets subtracts the value of the last payment. Notice that n increased to 6 with only 5 payments; therefore, the general formula for finding the future value of an annuity due is: (F/a,i,n)(17.10) To find the annuity due, given the future value, the formula would be: (a/f,i,n)(17.11) In order to find the present value of an annuity due of 5 payments use the cash flow diagram: The easiest approach is to find the future value of this annuity due using a modification of Equation (17.6): The present value of this payment series exists at the same point in time as the first payment. Modifying the cash flow diagram, we have: The cash flow diagram requires modifications as: Ignoring the first payment, the cash flow diagram appears as an ordinary annuity of 4 payments, applying Equation (17.8), we have: By inserting a payment (which does not exist) at the end of year 5, the cash flow diagram now looks like an ordinary annuity of 6 payments. Therefore, using Equation (17.6): This yields the present value of 4 of the 5 payments but does not consider the first payment, which is at time zero, or the same point as the present value. Therefore, the first payment, which is zero interest periods away from Pv, must be added. By modifying the above equation and placing a + 1 as the last value inside the brackets adds the value of the first payment and gives: 367

10 The general formula for finding the present value of an annuity due is: (P/a,i,n)(17.12) To find an annuity due, given the present value, the formula would be: (a/p,i,n)(17.13) Problems Involving Multiple Functions Many problems in cost analysis involve the use of several of the formulas presented thus far in this chapter. Example 17.15: Bonds. Bonds are sold in order to obtain investment capital. Most bonds pay interest on the face value (the value printed on the bond) either annually or semiannually, and pay back the face value at maturity (the end of the loan period, or end of the life of the bond). Consider a bond with a face value of $100,000, a life of 10 years, that provides annual interest payments of 8%. How much should be paid for this bond to make it yield 10%? Thus, the cash flow diagram becomes: Price = 8k(P/A,7,10) + 100k(P/F,7,10) Price = $56, $50, Price = $107, Example 17.16: Bank Loans. A couple financed $50,000 on a home. The terms of the home mortgage were 9.6% compounded monthly for 30 years. After making payments for 5 years, they want to calculate the amount of money they will pay to principal and interest during the 6th year. Step 1 is to calculate the amount of the monthly loan payment. Using Equation (17.9) gives: where Pv = $50,000 n = 360 i = 0.096/12 = or 0.8% A = 50,000(A/P,0.8,360) A = $ Now that the payment is calculated, Step 2 is to find the balance of the loan at the end of year 5. Using Equation (17.8) gives: where The above diagram illustrates the cash flows associated with the bond. The interest is paid out annually to the bond holder. Because there is no opportunity to earn interest on that interest, the bond performs as though it were a problem in simple interest. The performance of the bond cannot deviate from the original cash flow diagram; that is, the bond will always pay $8,000 per year plus $100,000 at maturity. To make this bond pay 10%, the buyer would have to pay less than $100,000. In other words, the bond would have to be sold at a discount price. To calculate the price to pay for the bond to make it yield 10%, the procedure would be: Price = 8k(P/A,10,10) + 100k(P/F,10,10) Price = $49, $38, Price = $87, To illustrate a premium paid for a bond, assume that the buyer was willing to purchase the bond to yield 7%. Therefore, to make the bond yield lower than 8%, the buyer would have to pay a premium. A = $ i = n = or 300 payments remaining to be made through year 30. Pv = (P/A,0.8,300) Pv = $48, Step 3 is to calculate the loan balance at the end of year 6, letting n = = 288 payments remaining to be made, and using the same process as in Step 2: Pv = (P/A,.8,288) Pv = $47, Subtracting the loan balance, end of year 6, from the loan balance, end of year 5, = $487.16, the amount that will be paid to principal during year 6. Step 4 is to calculate the amount of the payments that will go to interest. The total amount paid during year 6 will be $ x 12 or $5, Subtracting the amount that will go to principal, ($487.16), leaves the amount $4, that will go to interest during year

11 Example 17.17: Investments. A couple with two children, ages 2 and 4, want to invest a single annual payment series that will provide $10,000 to each child at the age of 18. The investment will earn interest at 8.75%. If the annual payments start today and the last deposit is made 16 years from today, what is the amount of each annual payment? To solve this problem, begin by drawing a cash flow diagram as: Pv = $2, $2, Pv = $5, This provides the total present value 1 year before the first payment. Now the payment series can be calculated as an ordinary annuity of 17 payments, using Equation (17.9) we have: The easiest approach is to find the future value of the required monies and then solve for the annuity. Although it is realized that the older child will withdraw $10, years from today, the equivalent value of that $10,000 can be evaluated at the end of the 16th year. Using Equation (17.2) gives: Fv = Pv(1 + i) n Fv = 10,000 ( ) ,000 Fv = $21, Then, using Equation (17.7), and making n = 17, the amount of each payment is: A = $ If this method is confusing, the problem could have been solved using present value. This requires more calculations and also modification of the cash flow diagram. To approach the problem from a present value point of view, draw a 17-year cash flow diagram as: In order to solve the problem using present value techniques, it is necessary to note that present value exists 1 year before the first payment. This is the reason for modification of the cash flow diagram. The first $10,000 payment must be discounted 15 years and the second $10,000 payment must be discounted 17 years. Using Equation (17.3) we have: A = $ Notice that, although the final answers are identical, it required much more effort and calculation to solve the problem from a present value point of view. It is advisable to spend some time evaluating problems, drawing cash flow diagrams and considering the simplest approach. Without a cash flow diagram, it would have been very easy to make the error of assuming there were only 16 payments. It is also doubtful that the problem solver could recognize that the easiest approach for this problem is to work with future value. If there is any doubt regarding the answer, it may be verified by working the problem backwards. Suppose annual payments of $ are deposited into an investment for 15 years. The balance in the account, using Equation (17.6) would be $17, At this time, the older child withdraws $10,000, leaving a balance of $7, This balance will earn interest for 2 more years, and using Equation (17.2), will amount to $8, Meanwhile, two more payments of $ will be made into the account. These two payments, with interest, will amount to $1,260.85, using Equation (17.6). The account balance, then, will amount to $9, The reason the answer is off by 3 cents is that the actual value of each payment was $ Using that value as the payment, the answer would equal exactly $10, COST COMPARISON OF INVESTMENT ALTERNATIVES For the most part, cost analysis involves selection of the minimum cost or maximum profit alternatives. There are basically four accepted methods of evaluating Alternative A as compared to Alternative B as compared to Alternative C. 369

12 Present Value Method The first of these methods is the present value technique, wherein all costs and revenues are discounted back to present value to arrive at a net present value for the project. This method is very time consuming if performed manually and presents problems when the alternatives have different economic lives. Alternative A may have an expected life of 10 years, Alternative B may have an expected life of 15 years, and Alternative C may have an expected life of 20 years. Therefore, in order to do any meaningful evaluation of these three alternatives in terms of present value, it is necessary to find a common denominator for their expected life, which, in this case, would be 60 years, where Alternative A would be replaced six times, B replaced four times, and C replaced three times Future Value Technique The future value technique of evaluating alternatives is almost identical to the present value method except that all costs and revenues are stated in terms of future value. The problem still arises of alternatives with incompatible useful lives. The above techniques are self-explanatory, and because they require such extensive calculations, they will not be covered further. But, because they exist, when evaluating alternatives using computers, it is convenient to indicate, somewhere in the output data generated, the net present value (NPV) of each alternative. Many organizations base their decisions on NPV, and governmental agencies, expect to evaluate benefit-cost ratios. The benefit-cost ratio is the ratio of benefits provided by the alternative versus cost incurred, and will be discussed later in this chapter. Evaluate a piece of equipment that costs $10,000, has a 6-year useful life with a salvage value of $2,000, and annual operating costs of $5,000. The interest rate used for the proposed evaluation is 9% compounded annually. To calculate the cost of purchasing this equipment, operating it for 6 years, and salvaging it at the end of 6 years for $2,000 at 9% interest, the procedure would be to draw a cash flow diagram as: The typical approach would be to find the annual cost of $10,000, subtract the annual cost of $2,000 salvage, and add the annual operating cost of $5,000. The annual equivalent of the purchase may be found by using Equation (17.9) as: Next, find the annual equivalent cost of the salvage value, using Equation (17.7), as: Annual Equivalent Cost Method The annual equivalent cost method of evaluating alternative projects states all costs and revenues over the useful life of the project in terms of an equal annual payment series (an ordinary annuity). This is probably the most widely used method in the industry, for several reasons: Finally, add the $5,000 operating cost. The total calculations appear as: A = $2, $ $5K 1. It requires less effort and fewer calculations. 2. It eliminates the problem of alternatives with incompatible useful lives. 3. It allows for much more sophistication when considering inflation, increasing equipment cost, equipment depreciation schedules, etc. The annual cost method assumes that the project will live forever, and that, if Alternative A has a useful life of 10 years and Alternative B has a useful life of 15 years, each alternative will be replaced at the end of its useful life. Therefore, the alternative with the minimum annual cost or maximum annual profit is the alternative that will be chosen. A = $6, The value of Equation (17.9), which was used to find the annual equivalent cost of equipment purchased is The value of Equation (17.7), which was used to find the annual equivalent cost of the salvage value, is

13 Notice that the difference between and is exactly equal to the interest rate (i), This is true for all interest rates, provided that these functions have the same i and the same n. In calculating the annual equivalent cost of purchasing the equipment and subtracting the annual equivalent cost of the equipment salvage at some time in the future, this is always the case: i and n are equal; and Equation (17.9) - i = Equation (17.7). Using functional notation as symbols for these formulas we have: [(A/P,9,6) ] = (A/F,9,6). Making this substitution, the annual cost formula becomes: A = (A/P,9,6)10K - [(A/P,9,6) ]2K + 5K multiplying, A = (A/P,9,6)10K - (A/P,9,6)2K (2K) + 5K factoring out (A/P,9,6), A = (A/P,9,6)(10K - 2K) (2K) +5K the general equation for the annual cost equation becomes: then A = (A/P,i,n)(cost - salvage) + i(salvage) + OC (17.14) represents the annual equivalent cost formula where OC = annual operating cost. This modification of the annual cost formula greatly reduces the amount of calculation necessary to arrive at the annual equivalent cost. Values can be produced by this formula as monthly equivalent costs or weekly equivalent costs by simply changing i to the interest rate per period and allowing n to equal the total number of periods. Using the previous example, suppose the annual percentage rate was 9% compounded monthly. To calculate the periodic costs in terms of monthly equivalent costs (as a monthly ordinary annuity): A = (A/P,0.75,72)(10K - 2K) (2K) + 5K/12 A = $ $ $ A = $ Stating costs as an ordinary annuity has nothing to do with actual cost flows. It simply states all costs and revenues as an equal payment series in order that one alternative may be compared with another (See Figure 17.2). Figure 17.2 Annual Equivalent Cost. 371

14 Rate of Return Method The rate of return (ROR) method of comparing alternatives calculates the interest rate for each alternative and selects the highest ROR. A word of caution is necessary, ROR evaluates INVESTED capital and the costs of operation and maintenance as opposed to revenues or benefits received from the project. Therefore, a project totally financed with borrowed money has no rate of return because there is no invested capital. The following examples assume 100% equity financing and are used to illustrate rate of return calculations. Example 17.18: An investment of $7,000 is placed into an account for a 10-year period. At the end of 10 years, the balance in the account is $17, What was the annual interest rate earned? Using Equation (17.2): To further simplify this problem, consider the differences between these two alternatives. If Alternative A is subtracted from Alternative B, the cash flow diagram becomes: Although the insurance premiums are an annuity due, the cash flow diagram above makes the cash flows appear as an investment of $1.5 providing an ordinary annuity of $1 at the end of each year for 2 years. The analysis has simplified the problem considerably and avoided using the more complex formulas associated with annuities due. Using Equation (17.8): Fv = Pv(1 + i) n $17, = $7,000(1 + i) 10. This problem can be solved by dividing both sides of the equation by $7,000 giving: 17,699.30/7,000 = (1 + i) 10 Entering values in this formula requires that the mathematical signs be properly observed. If money flowing out, below the time line, is considered positive (+) then money flowing in, above the time line, or money saved is considered negative (-). The calculation then becomes: = (1 + i) 10. Extracting the tenth root of each side of the equation gives: (2.528) 0.1 = [(1 + i) 10 ] 0.1 Notice there is one equation and one unknown (i), but the unknown appears three times in the equation. Therefore, the only solution would be by an iterative = 1 + i i = A ROR can also be calculated by evaluating the difference between alternatives that provide cost savings rather than revenues. Example 17.19: Fire insurance premiums on a warehouse are $500/y (Alternative A). The same coverage can be purchased by paying a 3-year premium of $1,250 (Alternative B). Find the ROR realized by purchasing a 3- year policy in place of three 1-year policies. To simplify this problem, notice that the 3-year insurance premium is 2.5 times the annual premium. The cash flow diagram for Alternative A appears below. process. Continuing with Equation (17.8): where I = 0.21, or 21% Pv = 1.5 A = - 1. The answer to this problem is , indicating that the first interest rate was too low. Try i = 22% The cash flow diagram for Alternative B appears below. The answer, , indicates the interest rate is too high. The actual interest rate is %. Placing this interest rate into the formula for i, the answer equals zero, indicating that the ROR realized by purchasing a 3-year 372

15 policy instead of three 1-year policies is 21.5%. There are many pocket calculators costing under $20 that are designed to calculate financial functions and are capable of iterating for problems involving a single financial function. Consider the annual cost formula where: a = (A/P,i,n) x ( cost - salvage) + (i) salvage + OC - revenues. Suppose it is required to find the ROR on such a project. Once again, the solution to the problem is found by iteration. But in this case, there is more than one financial function. Choose an interest rate. Work the problem at that interest rate, and find out if the answer comes out positive or negative. With the mathematical signs we have used in this example, if revenues exceed costs the answer would be negative; if costs exceed revenues the answer would be positive. When the exact i is placed into the formula that denotes the ROR, the answer would be zero. That is, the interest rate that causes revenues and costs to be exactly equal. Interest tables may be used to help find an upper and lower range for i to reduce the number of iterations necessary. A much easier solution is to place all of the cash flows by year into a computer and write a simple program that can do thousands of iterations in a matter of seconds to find i. Trying to iterate i for a problem with more than one function on a financial calculator nearly always results in error because of the tremendous amount of time and data that must be entered by hand. Even with a computer program, if the cash flows turn from negative to positive and back to negative during the project life, i can take on the form of a quadratic equation and the computer is unable to determine whether i is positive or negative. There are computer spreadsheets and other software available for iterating i, but these have their limitations. One of the most popular and widely used spread-sheets is designed to iterate i for a series of cash flows. However, this spreadsheet requires a rather accurate guess for i; because, if it does not find i after 20 iterations, the resultant answer is "Err" (error). The RELCOST program developed to accomplish LCC analysis (copyright, Washington State Energy Office) will iterate i to seven decimal places in a matter of seconds for projects with over 150 input variables and more than 500 inflation rates, and evaluate the iterated i to determine whether it is positive or negative. This program will be discussed in more detail in Subsection 17.7, entitled "LIFE CYCLE COST ANALYSIS." 17.5 GRADIENTS Arithmetic Gradients Having developed formulas for equal payment series involving both ordinary annuities and annuities due, an endof-period payment series that increases by a fixed dollar amount at the end of each period can now be evaluated. Consider the following end-of-period payments: As can be seen in the cash flow diagram above, the payment series started at $100 and increased by $20 per year. Such a payment series is called an arithmetic gradient. Although such a payment series is covered in detail by most texts on engineering economics, it is highly unlikely that any portion of a project will contain such a payment series. The method used to solve for present value, future value or annual equivalent cost of such a payment series is to break it up into a series of ordinary annuities. This payment series now appears as five ordinary annuities. Because the series is progressing in the direction of future value, it is easiest to assume one annual payment series of $100 and find the future value of the sum of the remaining four annuities and string them out in the form of an annual equivalent for the 5-year period. The formula to accomplish this, where n = 5 and i = 9%, is: The 5-year equivalent cost of an arithmetic gradient of $20 that begins at the end of year 2 and increases through year 5 is $ Therefore, the annual equivalent of the payment series that began with $100 and increased by $20 for the next 4 years is $100 + $36.56, or $ Notice that when dealing with an arithmetic gradient, A 1 becomes an ordinary annuity, and the amount of increase starting at the end of the second period is considered to be the gradient. The gradient formula is based on the fact that the gradient always begins at the end of the second period and is to be strung out as an equal payment series from the end of the first period to the end of the cash flow. The general formula for finding the equivalent annual cost of an arithmetic gradient is: (A/G,i,n)(17.15) This is the arithmetic gradient uniform series factor where G is the amount of increase beginning at the end of the second period. 373

16 If the present value or future value of an arithmetic gradient is required, one could simply multiply Equation (17.15) by (P/A,i,n) or (F/A,i,n). The most common use of arithmetic gradients is in calculating the value of sum-of-years-digits (S-Y-D) depreciation, which takes the form of a negative arithmetic gradient. This method of depreciation is no longer allowed under the 1987 tax law. Note that: where g is constant. Therefore: Geometric Gradients A geometric gradient is an end-of-period payment series that increases by a fixed percentage each period. Consider a 5-year payment series that begins with $100 and increases by 9% every year thereafter. A cash flow diagram used for calculation of the problem is shown as: Then, to find the annual equivalent cost (A), considering inflation: annual cost = present cost x capital recovery factor (A/P,i,n) and is written: A = PV x (A/P,i,n) The cash flow diagram above illustrates a geometric gradient with the first payment (A 1 ) equal to $100 and each subsequent payment increasing by 9% in other words, a $100 payment inflating at a rate (g) of 9% annually. Then: Multiplying by the capital recovery factor: and the present value (Pv) for an assumed cost of capital of 12% compounded annually (i) is: Simplifying: (A/A 1,g,i,n) (17.16) Let n equal the number of interest periods (years, in this case). Then: where g does not equal i. This is the geometric gradient uniform series factor, where g does not equal i. Using functional notation: 374

17 Inserting the initial values in this example: (0.2774) = $ The annual equivalent cost equals $ Figure 17.3, plots a geometric gradient increasing by 9% for 20 years and indicates the annual equivalent cost at a discount rate of 12%. Notice, by removing the capital recovery factor in Equation (17.16), the equation becomes: where g is not = i. (P/A,g,i,n)(17.17) This is the geometric gradient present worth factor, where g is not = i. In the case where the discount rate equals the rate of inflation (g = i), the equation becomes A 1 divided by zero, which is undefined. If a geometric gradient is increasing by exactly the discount rate, this has the same effect of an interest rate equal to zero; therefore, simply take A 1, multiply it by the total number of payments (n), which will yield a present value at the end of year one. Because present value represents the dollar equivalent at time zero, the amount that was calculated by multiplying A 1 by n is one interest period off. To bring it to the proper time frame, time zero, simply discount it by one time period. Therefore: where g = i. (P/A 1,g,i,n) (17.18) This is the geometric gradient present worth factor, where g = i. Although geometric gradients are rather common and are found in many applications, they require that the rate of increase remain constant. Such is not the case in most energy forecasts, where inflation rates are modified year by year. This problem will be discussed later in Section 17.7 under the heading, "LIFE CYCLE COST ANALYSIS." 20-year Annual Equivalent Costs Project life in years 20 Interest rate (APR) 12% Capital investment $120,000 $16,065 Salvage value 12, Annual costs Insurance Fixed 2,250 2,250 Arithmetic gradient 230 1,385 Geometric gradient increasing at 10% 2,000 4,051 increasing at 6% 10,000 14,895 Depreciation method 200% declining balance Figure 17.4 illustrates the total annual equivalent costs for operating this project for 1 year, 2 years, 3 years, etc. The capital recovery line indicates the annual equivalent of investing capital and salvaging the project at end of year 1, 2, 3, etc. The operation and maintenance line shows the annual equivalent cost of operating the project in years 1 through 20. Notice that the minimum annual cost occurs in year 15, which is $38,486. That is to say, the annual equivalent stream of equal payments for operating the project with a 15-year life would be $38,486/y. Table 17.2 Economic Life Calculations Capital O&M Total Salvage Year Recovery Costs Costs Values 1 $38,400 $14,810 $53,210 $96, ,777 15,296 50,073 76, ,754 15,779 47,533 61, ,224 16,259 45,483 49, ,100 16,735 43,834 39, ,311 17,207 42,517 31, ,800 17,674 41,473 25, ,519 18,135 40,655 20, ,431 18,592 40,023 16, ,554 19,042 39,596 12, ,629 19,485 39,114 12, ,875 19,922 38,797 12, ,253 20,352 38,605 12, ,734 20,774 38,509 12, ,297 21,189 38,486 12, ,926 21,596 38,522 12, ,609 21,995 38,604 12, ,337 22,385 38,722 12,000 Figure 17.4 graphs a project with the following input variables: 375

18 Actual cash flow 5 4 Annual equivalent Years Figure 17.3 Geometric gradient DOLLARS (Thousands) Total Annual Cost O & M 20 Capital Recovery YEARS Figure 17.4 Economic life. 376

19 Table 17.2 provides annual equivalent values from year 1 through year 18 for capital recovery, operation and maintenance costs, and total annual costs. Salvage values for any given year are indicated in the last column. This analysis is beyond the scope of this chapter, but could be used to determine the economic life (minimum annual cost) of a project that had these cost characteristics. Table 17.2 provides answers for various costs that could be beneficial to those who want to sharpen their expertise in calculating capital recovery, annuities due, ordinary annuities, geometric gradients, and arithmetic gradients. in this case, 2. This sum is then divided into cost minus salvage to obtain one unit of depreciation, which is: Example: Equipment cost = $60,000 Salvage value = $ 5,000 then, 17.6 EQUIPMENT DEPRECIATION A discussion of equipment depreciation is essential in evaluating projects for taxable entities because equipment depreciation significantly lowers the annual cost of a project. This subject is also the area of constant change because it changes as tax laws are revised. The amount of total capital investment in a project and the reduction in tax liability caused by depreciation or investment and energy tax credits or both, all have a major bearing on whether or not the project is economically feasible. However, the 1986 tax law drastically reduced many of these incentives. Competent tax accountants should be a part of the development team to ensure proper utilization of these considerations Straight Line Depreciation Straight line depreciation is the simplest form of depreciation and has survived tax law changes for many decades and it is accepted by the 1987 tax law. The formula for straight line depreciation is: 3. This unit of depreciation is then multiplied by the years of life in descending order, that is: year 1 = 10 units = $10,000 year 2 = 9 units = $ 9,000 year 3 = 8 units = $ 8, year 10 = 1 unit = $ 1,000. The depreciation charge under this method performs like a negative arithmetic gradient where A 1 is $10,000 and G is -$1,000. Although this method of depreciation is accepted accounting practice and may be used on equipment purchased before 1980, it is no longer allowed under the 1987 tax law and is only discussed to illustrate an application of the arithmetic gradient Declining Balance Depreciation Life can be expressed in years, months, units of production, operating hours, or miles. Under the 1987 tax law, salvage values are set to zero Sum-of-Years Digits Depreciation This depreciation method is an accelerated depreciation schedule that recovers larger amounts of depreciation in the early life of the asset. The method of calculation is as follows: 1. Find the sum of the years' digits of the life of the equipment. Example: For a 10-year life, the sum of the digits 1 through 10 is equal to 55. The easiest way to make this calculation is: This method of depreciation is also an accelerated form and may obtain even larger depreciation in the early life than S-Y-D. With 200% declining balance depreciation, the annual rate is 200% times the straight line rate. As an example: An asset with a 10-year life has a straight line rate of 1/10. Therefore, the 200% declining balance rate would be 2/10 or 20%. This rate is applied to the book value of the asset, where book value equals cost minus accumulated depreciation. Any salvage value of the equipment was not considered except that the tax code provided that the equipment could not be depreciated below its salvage value. The $60,000 piece of equipment in the example above would be evaluated using 200% declining balance as shown in Table

20 Table 17.3 Declining Balance Depreciation Using 200% Declining Balance Book Value x Annual Book Value Depreciation Rate Depreciation End of Year Year ($) ($) ($) 1 60,000 x ,000 48, ,000 x ,600 38, ,400 x ,680 30, Although this method of depreciation provides the maximum write-off in the early life of the equipment, the annual depreciation charge rapidly decreases to the point that it would be beneficial to switch to straight line after the 6th year. It is interesting to note that the declining balance method of depreciation, whether it be 200%, 150% or 125%, takes the form of a negative geometric gradient and the book value for any year can be calculated using Equation (17.2). In this example, Fv = $60,000 (1-0.20) 3 Fv = $30,720 = book value end of year 3, to calculate the depreciation for year 6, Fv = [60,000 ( )] 0.20 Fv = $3,932,16. In the example above, the book value is calculated for the end of year 5, and multiplied by the depreciation rate, 0.20, to obtain the annual depreciation for year 6. Once again, although this method of depreciation is an accepted accounting method, declining balance depreciation was made obsolete with the 1980 tax law changes. However, a modified version of this method was reinstated with the 1986 tax code revision and is discussed below Modified Accelerated Cost Recovery System (MACRS) The Accelerated Cost Recovery System (ACRS) was introduced in 1981, but underwent major revision in 1986, effective with the 1987 tax year. In 1989, the Modified Accelerated Cost Recovery System (MACRS) was introduced, which was another major revision. MACRS depreciation is designed to provide rapid depreciation and to eliminate disputes over depreciation methods, useful life, and salvage value. The depreciation method and useful life are fixed by law, and the salvage value is treated as zero. MACRS depreciation rates depend on the recovery period for the property and whether the mid-year or mid-quarter convention applies. Property with class lives of 3, 5, 7, and 10 years may be depreciated at either the 200% or the 150% declining balance rate, with a switch to straight line. Property with class lives of 15 and 20 years must be depreciated using the 150% declining balance rate with a switch to straight line. In either case, the switch to straight line occurs when the straight line rate provides larger annual deductions. Table 17.4 lists the various classes of depreciable property under the Modified Accelerated Cost Recovery System. Table 17.4 MACRS Class Lives Property All personal property other than real estate. Class Special handling devices used in the manufacturing of food and beverages. 3-Year Property Special tools and devices used in the manufacture of rubber products, fabricated metal products, or motor vehicles, and finished plastic products. Property with a class life of 4 years or less. Automobiles, light-duty trucks (unloaded weight of less than 13,000 pounds). Semi-conductor manufacturing equipment. Aircraft owned by non-air transport companies Typewriters, copiers, duplicating equipment, heavy general purpose trucks, trailers, cargo containers, and trailer mounted containers. Computers. 5-Year Computer-based telephone central office Property switching equipment, computer-related Peripheral equipment, and property used in research and experimentation. Equipment qualifying as a small power production facility within the meaning of Section 3(17)(C) of the Federal Power Act (16 U.S.C. 796 (17)(C)), as in effect on Petroleum drilling equipment. Property with a class life of more than 4 and less than 10 years. 378